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Smooth cubic surfaces with 15 lines

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Abstract

In this paper, we consider smooth cubic surfaces with 15 lines. It is known that such surfaces can be generated by means of a double six with two pairs of Galois conjugate lines defined over the quadratic extension. The approach taken here is to consider the generation by means of a set of 9 lines defined over the field of coordinates. Eight lines arise from the double six, while the ninth is the diagonal line of the two pairs of Galois conjugate lines. This allows us to express all necessary equations and objects in terms of a set of four parameters over the coordinate field. As an application, we classify the smooth cubic surfaces with 15 lines over small finite fields by computer. All our results match with an enumerative formula recently found by Das.

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Acknowledgements

I would like to thank Anton Betten for constructive criticism of the manuscript. I also like to thank Math department at Colorado State University for giving me permission to use their server called ’Otto’. I would also like to thank the two referees for their many comments and suggestions, which helped improve the paper quite a bit. This work is supported by “The Scientific and Technological Research Council of Turkey” (Grant Number: 1059B192000479).

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  1. Department of Mathematics, Gebze Technical University, Kocaeli, Turkey

    Fatma Karaoglu

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  1. Fatma Karaoglu
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Correspondence to Fatma Karaoglu.

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Karaoglu, F. Smooth cubic surfaces with 15 lines. AAECC 33, 823–853 (2022). https://doi.org/10.1007/s00200-022-00582-3

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